import numpy as np

def eliminateCol$($A$,$ k$=0)$:

# guarantee that A is float type

if A$.$dtype.kind $!=\text{'}$f$\text{'}$ and A$.$dtype.kind $!=\text{'}$c$\text{'}$: return None

# load dimensions

dim$0,$dim$1=$ np$.$shape$($A$)$

# initialize outputs

swap $=$ np$.$identity$($dim$0,$dtype$=$float$)$

oper $=$ np$.$identity$($dim$0,$dtype$=$float$)$

# k $<$ dim$1$

k $=$ k$\%$dim$1$

# if the column k is zero column, return

if A$[$:$,$k$].$any$()==$ False: return swap,oper

i$0=0$

if k $!=0$:

# find a row i$0$ whose leftmost k$-1$ entries are all zero

while i$0<$ dim$0$:

if not A$[$i$0,0$:k$].$any$()$:

break

i$0+=1$

# find a nonzero element A$[$i$,$ k$]$ to use as a pivot

i $=$ i$0$

while i $<$ dim$0$:

if A$[$i$,$ k$]!=0$:

break

i $+=1$

if i $==$ dim$0$:

return swap, oper

# Partial Pivoting: ensures that the largest element of the column is placed on the diagonal

max$\_$index $=$ np$.$argmax$($np$.$abs$($A$[$i$0$:$,$ k$]))+$ i$0$

A$[[$i$0,$ max$\_$index$],]=$ A$[[$max$\_$index, i$0],]$

swap$[[$i$0,$ max$\_$index$],]=$ swap$[[$max$\_$index, i$0],]$

# row i$0$ is the pivot row and A$[$j$,$ k$]$ is zero for i$0<$ j $<=$ i

i $=$ i $+1$

while i $<$ dim$0$:

if A$[$i$,$ k$]!=0$:

quot $=$ A$[$i$,$ k$]/$ A$[$i$0,$ k$]$

A$[$i$,$ :$]=$ A$[$i$,$ :$]-$ quot $*$ A$[$i$0,$ :$]$

oper$[$i$,$ i$0]=-$quot

i $+=1$

return swap, oper

def eliminateRow$($A$,$ k$=-1)$:

# guarantee that A is float type

if A$.$dtype.kind $!=\text{'}$f$\text{'}$ and A$.$dtype.kind $!=\text{'}$c$\text{'}$: return None

# load dimensions

dim$0,$dim$1=$ np$.$shape$($A$)$

# initialize outputs

oper $=$ np$.$identity$($dim$0,$dtype$=$float$)$

# k $<$ dim$0$

k $=$ k$\%$dim$0$

# if the row k is zero column, return

if k $==0$ or A$[$k$,$:$].$any$()==$ False: return oper

# find a nonzero element A$[$k$,$j$]$ to use as a pivot

j $=0$;

while j $<$ dim$1$:

if A$[$k$,$j$]!=0$: break

j$+=1$

i $=$k$-1$

while i $>=0$:

if A$[$i$,$j$]!=0$:

quot $=$ A$[$i$,$j$]/$ A$[$k$,$j$]$

A$[$i$,$:$]=$ A$[$i$,$:$]-$ quot$*$A$[$k$,$:$]$

oper$[$i$,$k$]=-$quot

i$-=1$

return oper

def eliminateScale$($A$)$:

# guarantee that A is float type

if A$.$dtype.kind $!=\text{'}$f$\text{'}$ and A$.$dtype.kind $!=\text{'}$c$\text{'}$: return None

# load dimensions

dim$0,$dim$1=$ np$.$shape$($A$)$

# initialize outputs

oper $=$ np$.$identity$($dim$0,$dtype$=$float$)$

# find pivots

i $=0$

while i $<$ dim$0$:

j $=0$

while j $<$ dim$1$:

if A$[$i$,$j$]==0$: j$+=1$; continue

else:

oper$[$i$,$i$]=1/$A$[$i$,$j$]$

A$[$i$,$j:$]=$ A$[$i$,$j:$]/$A$[$i$,$j$]$

break

j$+=1$

i$+=1$

return oper

def get$\_$echelon$\_$form$($A$)$:

# Perform Gaussian elimination on matrix A to achieve Row Echelon Form

for k in range$($min$($A$.$shape$[0],$ A$.$shape$[1]))$:

# Find the k$-$th pivot

i$\_$max $=$ max$($range$($k$,$ A$.$shape$[0]),$ key$=$lambda i: abs$($A$[$i$,$ k$]))$

if A$[$i$\_$max, k$]==0$:

continue # Skip if the column is zero

# Swap rows

A$[[$k$,$ i$\_$max$]]=$ A$[[$i$\_$max, k$]]$

# Make all rows below this one $0$ in the current column

for i in range$($k $+1,$ A$.$shape$[0])$:

f $=$ A$[$i$,$ k$]/$ A$[$k$,$ k$]$

A$[$i$,$ k:$]-=$ f $*$ A$[$k$,$ k:$]$

return A

def get$\_$reduced$\_$echelon$\_$form $($A$)$:

# Perform Gaussian elimination on matrix A

for k in range$($A$.$shape$[1])$:

eliminateCol$($A$,$ k$)$

eliminateRow$($A$,$ k$)$

# Finally, scale the matrix

eliminateScale$($A$)$

return A

# Example Usage:

C $=$ np$.$array$([[0\mathrm{.}02,0\mathrm{.}01,0\mathrm{.}0,0\mathrm{.}0,0\mathrm{.}02],$

$[1\mathrm{.}0,2\mathrm{.}0,1\mathrm{.}0,0\mathrm{.}0,1\mathrm{.}0],$

$[0\mathrm{.}0,1\mathrm{.}0,2\mathrm{.}0,1\mathrm{.}0,4\mathrm{.}0],$

$[0\mathrm{.}0,0\mathrm{.}0,100\mathrm{.}0,200\mathrm{.}0,800\mathrm{.}0]],$ dtype$=$float$)$

B $=$ np$.$array$([[1\mathrm{.}0,3\mathrm{.}0,2\mathrm{.}0,5\mathrm{.}0],$

$[2\mathrm{.}0,4\mathrm{.}0,-6\mathrm{.}0,-4\mathrm{.}0],$

$[1\mathrm{.}0,5\mathrm{.}0,3\mathrm{.}0,10\mathrm{.}0]],$ dtype$=$float$)$

A $=$ np$.$array$([[1\mathrm{.}0,-1\mathrm{.}0,2\mathrm{.}0,8\mathrm{.}0],$

$[0\mathrm{.}0,0\mathrm{.}0,-1\mathrm{.}0,-11\mathrm{.}0],$

$[0\mathrm{.}0,2\mathrm{.}0,-1\mathrm{.}0,-3\mathrm{.}0]],$ dtype$=$float$)$

echelon $=$ get$\_$echelon$\_$form$($A$.$copy$())$

reduced $=$ get$\_$reduced$\_$echelon$\_$form$($A$.$copy$())$

print$($A$)$

print$($echelon$)$

print$($reduced$)$